We all learned in school about "special" right triangles. Special right triangles have integer side lengths. Examples include the $3$-$4$-$5$ right triangle, the $5$-$12$-$13$ right triangle, the $8$-$15$-$17$ right triangle, and their scalar multiples ($6$-$8$-$10$, $10$-$24$-$26$, $16$-$30$-$34$, etc).
How many are there? Is there a limit to the number of lowest-form (no scalar multiple) special right triangles? Are there any patterns that arise from the progression of integer side lengths?
Best Answer
Choose your favorite positive integers $m$ and $n$ with $m>n$. Then set $a=m^2-n^2$, $b=2mn$ and $c=m^2+n^2$. You will see then that we have the equation $a^2+b^2=c^2$.
Now, it just so happens that if you choose $m$ and $n$ so that they share no common factors, and such that $m-n$ is odd, then $a,b$, and $c$ also share no common factors. Can you see why this is true? (In fact, $a$ and $b$ won't even share any common factors themselves).
It turns out that every possible lowest-form triangle is derived from some appropriate choice of $m$ and $n$.
This answers your first question: there are infinitely many 'lowest form' triangles (called primitive triplets). It also partially answers your second question: the integer side length of one of the legs of a primitive special triangle is always the difference of two squares.